A conditional Lagrangian clock barrier at the C1,1/3 threshold for axisymmetric Euler without swirl
Abstract
We consider axisymmetric no-swirl solutions to the three-dimensional incompressible Euler equations, with initial velocity in C1,α∩ L2, where α∈[1/3,1). In a major breakthrough, Shkoller introduced a clock-and-driver framework that he used in order to prove finite-time type I blow-up below the C1,1/3 threshold in this setting. Motivated by this, we define Lagrangian classes of coherent conditional solutions for which the same mechanism yields a supercritical-critical barrier to blow-up when α≥1/3. When α>1/3, the aforementioned barrier is genuinely depleted, whereas at the critical endpoint α=1/3, we obtain an exponential bound preventing blow-up. In the general case, we formulate a matrix-clock criterion in terms of the smallest singular value of the deformation gradient and show that, under transverse cusp-tail, longitudinal, off-clock, Dini, and suitable geometric coherence hypotheses, this singular value cannot collapse in finite time. In particular, we also show that the class of such coherent solutions includes the smooth ones locally in time. In the on-axis case, the criterion reduces to the scalar clock inequality J̇(t)≳ −B(t)J(t)−CJ(t)3α, which rules out Shkoller-type clock collapse for α≥1/3. These results do not enlarge the known Lorentz-space global regularity classes. Rather, they in particular identify the supercritical Lagrangian obstruction dual to Shkoller's subcritical blow-up mechanism in the case α>1/3.
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BibTeX
@misc{avadanei2026conditional,
title = {{A conditional Lagrangian clock barrier at the $C^1,\frac13$ threshold for axisymmetric Euler without swirl}},
author = {Ovidiu-Neculai Avadanei},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.31587},
eprint = {2605.31587},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.31587},
}
RIS
TY - UNPB
TI - A conditional Lagrangian clock barrier at the C^1,1/3 threshold for axisymmetric Euler without swirl
AU - Ovidiu-Neculai Avadanei
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2605.31587
UR - https://arxiv.org/abs/2605.31587
AB - We consider axisymmetric no-swirl solutions to the three-dimensional incompressible Euler equations, with initial velocity in C^1,α∩ L^2, where α∈[1/3,1). In a major breakthrough, Shkoller introduced a clock-and-driver framework that he used in order to prove finite-time type I blow-up below the C^1,1/3 threshold in this setting. Motivated by this, we define Lagrangian classes of coherent conditional solutions for which the same mechanism yields a supercritical-critical barrier to blow-up when α≥1/3. When α>1/3, the aforementioned barrier is genuinely depleted, whereas at the critical endpoint α=1/3, we obtain an exponential bound preventing blow-up. In the general case, we formulate a matrix-clock criterion in terms of the smallest singular value of the deformation gradient and show that, under transverse cusp-tail, longitudinal, off-clock, Dini, and suitable geometric coherence hypotheses, this singular value cannot collapse in finite time. In particular, we also show that the class of such coherent solutions includes the smooth ones locally in time. In the on-axis case, the criterion reduces to the scalar clock inequality J̇(t)≳ −B(t)J(t)−CJ(t)^3α, which rules out Shkoller-type clock collapse for α≥1/3. These results do not enlarge the known Lorentz-space global regularity classes. Rather, they in particular identify the supercritical Lagrangian obstruction dual to Shkoller's subcritical blow-up mechanism in the case α>1/3.
ER -